Page 107 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

92 W. KUTZELNIGG

The series (A.4) has here the radius of convergence but it can be continued
analytically beyond its radius of convergence.
Let us now argue that we are actually interested in the integral

and that the first approximation step is to replace by y and the second one the
discretization, then the total error consists of the cut-off-error

and the discretization error (B.4).
The limit of the discretization error (B.4) is

while from the Fourier expansion (A.9) we get

The identity between (B.8) and (B.9) is not immediately recognized. One sees at
least easily that for small h one gets from (B.9)

in agreement with what one gets from the Taylor expansion of (B.8) or immediately
from (A.4). The agreement of (B.8) and (B.9) is confirmed in terms of a relation
familiar in the theory of the digamma function

together with